I’m uncertain how you would be forwarding text messages from your Republic phone to your Verizon phone. To the best of my knowledge, such a thing is not possible.
Presuming your Verizon phone is Android or iOS, Republic Anywhere will allow you to send and receive messages sent to your Republic number on the Verizon phone today. Calling for Anywhere is in beta but not yet ready for public release. More on Anywhere here: Texting App | Text from Computer | Republic Wireless.
Calls to your Republic number can be sent to your Verizon phone via voicemail forwarding. More on that here: How to Forward Your Voicemail Messages – Republic Help. Voicemail forwarding is essentially call forwarding upon no answer. If you leave your Republic phone off, calls will forward to your Verizon phone immediately.
If you need to make calls showing your Republic number from your Verizon phone, there are Voice over Internet Protocol (VoIP) solutions that allow for spoofing Caller ID, so that calls made would appear to be coming from your Republic phone. Caller ID spoofing is often considered (and can be) a bad thing. There are, however, perfectly legitimate reasons for employing it and there is nothing wrong with doing so for legitimate purposes. The rule of thumb is is one controls the number being spoofed, it’s legitimate to do so.